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- Nested_dissection abstract "In numerical analysis, nested dissection is a divide and conquer heuristic for the solution of sparse symmetric systems of linear equations based on graph partitioning. Nested dissection was introduced by George (1973); the name was suggested by Garrett Birkhoff.Nested dissection consists of the following steps:Form an undirected graph in which the vertices represent rows and columns of the system of linear equations, and an edge represents a nonzero entry in the sparse matrix representing the system.Recursively partition the graph into subgraphs using separators, small subsets of vertices the removal of which allows the graph to be partitioned into subgraphs with at most a constant fraction of the number of vertices.Perform Cholesky decomposition (a variant of Gaussian elimination for symmetric matrices), ordering the elimination of the variables by the recursive structure of the partition: each of the two subgraphs formed by removing the separator is eliminated first, and then the separator vertices are eliminated.As a consequence of this algorithm, the fill-in (the set of nonzero matrix entries created in the Cholesky decomposition that are not part of the input matrix structure) is limited to at most the square of the separator size at each level of the recursive partition. In particular, for planar graphs (frequently arising in the solution of sparse linear systems derived from two-dimensional finite element method meshes) the resulting matrix has O(n log n) nonzeros, due to the planar separator theorem guaranteeing separators of size O(√n). For arbitrary graphs there is a nested dissection that guarantees fill-in within a logarithmic factor of optimal, but this method is not guaranteed to achieve optimal fill-in and finding the optimal dissection is not a solved problem.".
- Nested_dissection wikiPageExternalLink 6607.
- Nested_dissection wikiPageID "28070753".
- Nested_dissection wikiPageLength "3815".
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- Nested_dissection wikiPageRevisionID "607168273".
- Nested_dissection wikiPageWikiLink Category:Numerical_linear_algebra.
- Nested_dissection wikiPageWikiLink Category:Sparse_matrices.
- Nested_dissection wikiPageWikiLink Cholesky_decomposition.
- Nested_dissection wikiPageWikiLink Cycle_rank.
- Nested_dissection wikiPageWikiLink Divide_and_conquer_algorithms.
- Nested_dissection wikiPageWikiLink Finite_element_method.
- Nested_dissection wikiPageWikiLink Garrett_Birkhoff.
- Nested_dissection wikiPageWikiLink Gaussian_elimination.
- Nested_dissection wikiPageWikiLink Glossary_of_graph_theory.
- Nested_dissection wikiPageWikiLink Graph_(discrete_mathematics).
- Nested_dissection wikiPageWikiLink Graph_partition.
- Nested_dissection wikiPageWikiLink Heuristic.
- Nested_dissection wikiPageWikiLink Numerical_analysis.
- Nested_dissection wikiPageWikiLink Planar_separator_theorem.
- Nested_dissection wikiPageWikiLink Recursion.
- Nested_dissection wikiPageWikiLink Sparse_matrix.
- Nested_dissection wikiPageWikiLink Symmetric_matrix.
- Nested_dissection wikiPageWikiLink System_of_linear_equations.
- Nested_dissection wikiPageWikiLink Vertex_separator.
- Nested_dissection wikiPageWikiLinkText "Nested dissection".
- Nested_dissection wikiPageWikiLinkText "nested dissection".
- Nested_dissection wikiPageUsesTemplate Template:Citation.
- Nested_dissection wikiPageUsesTemplate Template:Harvtxt.
- Nested_dissection wikiPageUsesTemplate Template:Reflist.
- Nested_dissection subject Category:Numerical_linear_algebra.
- Nested_dissection subject Category:Sparse_matrices.
- Nested_dissection hypernym Divide.
- Nested_dissection type Combinatoric.
- Nested_dissection type Matrix.
- Nested_dissection comment "In numerical analysis, nested dissection is a divide and conquer heuristic for the solution of sparse symmetric systems of linear equations based on graph partitioning.".
- Nested_dissection label "Nested dissection".
- Nested_dissection sameAs Q6997812.
- Nested_dissection sameAs m.0cmb9gf.
- Nested_dissection sameAs Q6997812.
- Nested_dissection wasDerivedFrom Nested_dissection?oldid=607168273.
- Nested_dissection isPrimaryTopicOf Nested_dissection.